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For a real smooth solution of the long-wave convection equation with broken Boussinesq symmetry, assume a periodic spatial average or an existing long-interval average with vanishing endpoint fluxes. Integration by parts gives the exact energy method identityThus $\mu <{\mu }_E$ prevents growth of the mean-square temperature at arbitrary amplitude in this averaging class. This sufficient nonlinear bound need not equal the linear instability threshold, and it does not imply pointwise monotonicity of the temperature.

For the long-wave convection equation with broken Boussinesq symmetry, use a sufficiently smooth real temperature field. To make the spatial average and integrations meaningful, take a periodic pattern, or an existing long-interval average with bounded derivatives and vanishing averaged endpoint fluxes. Boundedness of the temperature alone does not guarantee all those averaging properties. Multiply the evolution equation by $\Theta$ and average. Integration by parts givesand $⟨\Theta ({\Theta }_x^j{)}_x⟩=-⟨{\Theta }_x^{j+1}⟩$. Thus the energy method yieldsThe energy square completion for long-wave convection starts fromPut $v={\Theta }_x$. The energy identity becomesSince $sv-v^2=s^2/4-(v-s/2{)}^2\le s^2/4$ pointwise,Therefore $\mu <2-s^2/4$ excludes growth of the mean-square temperature, for arbitrary amplitude within this smooth averaging class. This is a nonlinear energy-stability criterion, not a proof of pointwise monotonicity at each position. A spatially constant component instead decays through the $-\Theta$ term. The criterion is sufficient; it need not coincide with the linear instability threshold.

In a weakly nonlinear expansion of the long-wave convection equation with broken Boussinesq symmetry at $\mu =2$, a critical Fourier mode ${\Theta }_1=A{e}^{ix}+c.c.$ generates ${\Theta }_2=(2is/9)A^2{e}^{2ix}+c.c.$. The quadratic interaction of these first and second harmonics feeds back $8s^2|A{|}^{2}A/9$ into the critical Fourier mode, while the cubic gradient nonlinearity contributes $-3|A{|}^{2}A$. The Fredholm solvability condition is consequently ${\mu }_{2}A-g|A{|}^{2}A=0$. The sign of $g$ distinguishes supercritical and subcritical branches; $g=0$ requires a higher-order amplitude equation.